The elliptic curve method (ECM) for integer factorization is an algorithm that uses the algebraic structure of the set of points of an elliptic curve for factoring integers. The running time of ECM depends on the size of the smallest prime divisor of the number to be factored. One of its main applications is the co-factorization step in the number field sieve algorithm that is used for assessin...

In this talk I review the structure of vacua of N = 2 theories broken down to N = 1 and it’s link with factorization of SeibergWitten curves. After an introduction to the structure of vacua in various supersymmetric gauge theories, I discuss the use of the exact factorization solution to identify different dual descriptions of the same physics and to count the number of connected domains in the...

The issue of factorization within the context of coincidence quasi–elastic electron scattering is reviewed. Using a relativistic formalism for the entire reaction mechanism and restricting ourselves to the case of plane waves for the outgoing proton, we discuss the meaning of factorization in the cross section and the role of the small components of the bound nucleon wave function. PACS number(...

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200000, the largest number factored to date using a quantum processor. We also explain how Gröbner bases can be used ...

Recommender systems are most often approached from a number-crunching perspective, even though recent algorithms are very similar to methods used in psychology. Based on these similarities this study tried to investigate if the prediction models produced by a matrix factorization algorithm can be interpreted like is done in more psychological studies. Multidimensional scaling was used on data g...

Abstract We provide elementary explanations in terms of palindromic continued fraction expansions for the factorization of integers of the form a2 + 1, including Fermat numbers and Cunningham project numbers. This provides a generalization and more complete explanation of the factorization of the sixth Fermat number given by Freeman Dyson at the turn of the century. This explanation may provide...

The security of the most popular asymmetric cryptographic scheme RSA depends on the hardness of factoring large numbers. The best known method for factorization large integers is the General Number Field Sieve (GNFS). One important step within the GNFS is the factorization of midsize numbers for smoothness testing, an efficient algorithm for which is the Elliptic Curve Method (ECM). We present ...

Let q = ps be a power of a prime number p and let Fq be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over Fq . In particular we obtain the explicit factorization of the cyclotomic polynomial Φ...

In this article, we factor the composite $4n^2 +1$ using Fermat's factorization method. Consequently, characterized all proper factors of in terms its form. Furthermore, number is considered study.

The number field sieve is a relatively new method to factor large integers. Its most notable success is the factorization of the ninth Fermat number. It is significantly faster than all known existing integer factoring algorithms. We examine the theoretical underpinnings of the sieve; after understanding how it works, we state the algorithm. We look mostly to the algebraic number theory aspects...